Abstract. Let $f(\varphi)$ be a positive continuous function on
$0\leq\varphi\leq\Theta$ , where $\Theta\leq2\pi$ , and let $\xi$
be the number of two-dimensional lattice points in the domain
$\Pi_R(f)$ between the curves $r=(R+c_1/R)f(\varphi)$ and
$r=(R+c_2/R)f(\varphi)$ , where $c_1<c_2$ are fixed. Randomizing
the function $f$ according to a probability law $\bold P$ , and
the parameter
$R$ according to the uniform distribution $\mu_L$ on the interval
$[a_1L,a_2L]$ , Sinai showed that the distribution of $\xi$ under
$\bold P\times\mu_L$ converges to a mixture of the
Poisson distributions as $L\to\infty$ .
Later Major showed that for $\bold P$-almost all $f$ ,
the distribution of $\xi$
under $\mu_L$ converges to a Poisson distribution as $L\to\infty$ .
In this note, we shall
give shorter and more transparent proofs to these interesting theorems,
at the same time extending the
class of $\bold P$ and strengthening the statement of Sinai.